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What is the correct way of expressing this probability mass function?

Suppose I have a discrete random variable $S$ which is discrete. Now further suppose that I have a random variable $X$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $P[S=s]$ for $S$ using the expression: \begin{equation} P[S=s]= \int_{\mathbb{R}}P[S=s\mid x]P[x\mid s]dx \end{equation}

A professor I spoke to mentioned that if both $S$ and $X$ were sicretediscrete, this expression would have been correct, however, given that $X$ is a continuous random variable, the way I have expressed it implies a zero probability (I suppose because of $P[x\mid s]$). I was told to express this correctly, I have to express it in terms of probability measures.

I have been spending the better part of yesterday on Shiryaev's probability book and on probability measures and Lebesgue integrals; but while I know he is correct about my incorrect expression, I am not sure exactly how this needs to be expressed in terms of probability measures.

What is the correct way of expressing this probability mass function

Suppose I have a discrete random variable $S$ which is discrete. Now further suppose that I have a random variable $X$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $P[S=s]$ for $S$ using the expression: \begin{equation} P[S=s]= \int_{\mathbb{R}}P[S=s\mid x]P[x\mid s]dx \end{equation}

A professor I spoke to mentioned that if both $S$ and $X$ were sicrete, this expression would have been correct, however, given that $X$ is a continuous random variable, the way I have expressed it implies a zero probability (I suppose because of $P[x\mid s]$). I was told to express this correctly, I have to express it in terms of probability measures.

I have been spending the better part of yesterday on Shiryaev's probability book and on probability measures and Lebesgue integrals; but while I know he is correct about my incorrect expression, I am not sure exactly how this needs to be expressed in terms of probability measures.

What is the correct way of expressing this probability mass function?

Suppose I have a discrete random variable $S$. Now further suppose that I have a random variable $X$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $P[S=s]$ for $S$ using the expression: \begin{equation} P[S=s]= \int_{\mathbb{R}}P[S=s\mid x]P[x\mid s]dx \end{equation}

A professor I spoke to mentioned that if both $S$ and $X$ were discrete, this expression would have been correct, however, given that $X$ is a continuous random variable, the way I have expressed it implies a zero probability (I suppose because of $P[x\mid s]$). I was told to express this correctly, I have to express it in terms of probability measures.

I have been spending the better part of yesterday on Shiryaev's probability book and on probability measures and Lebesgue integrals; but while I know he is correct about my incorrect expression, I am not sure exactly how this needs to be expressed in terms of probability measures.

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Carl
Carl
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What is the correct way of expressing this probability mass function

Suppose I have a discrete random variable $S$ which is discrete. Now further suppose that I have a random variable $X$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $P[S=s]$ for $S$ using the expression: \begin{equation} P[S=s]= \int_{\mathbb{R}}P[S=s\mid x]P[x\mid s]dx \end{equation}

A professor I spoke to mentioned that if both $S$ and $X$ were sicrete, this expression would have been correct, however, given that $X$ is a continuous random variable, the way I have expressed it implies a zero probability (I suppose because of $P[x\mid s]$). I was told to express this correctly, I have to express it in terms of probability measures.

I have been spending the better part of yesterday on Shiryaev's probability book and on probability measures and Lebesgue integrals; but while I know he is correct about my incorrect expression, I am not sure exactly how this needs to be expressed in terms of probability measures.